Abstract group 56882.18: $F_{239}$

• Label: 56882.18
• Order: \( 2 \cdot 7 \cdot 17 \cdot 239 \)
• Exponent: \( 2 \cdot 7 \cdot 17 \cdot 239 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 7 \cdot 17 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2 \cdot 7 \cdot 17 \cdot 239 \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $239$
• Trans deg.: $239$
• Rank: $2$

Group information

Description:	$F_{239}$
Order:	\(56882\)\(\medspace = 2 \cdot 7 \cdot 17 \cdot 239 \)
Exponent:	\(56882\)\(\medspace = 2 \cdot 7 \cdot 17 \cdot 239 \)
Automorphism group:	$F_{239}$, of order \(56882\)\(\medspace = 2 \cdot 7 \cdot 17 \cdot 239 \)
Composition factors:	$C_2$, $C_7$, $C_{17}$, $C_{239}$
Derived length:	$2$

This group is nonabelian and a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group).

Group statistics

Order	1	2	7	14	17	34	119	238	239
Elements	1	239	1434	1434	3824	3824	22944	22944	238	56882
Conjugacy classes	1	1	6	6	16	16	96	96	1	239
Divisions	1	1	1	1	1	1	1	1	1	9
Autjugacy classes	1	1	6	6	16	16	96	96	1	239

Dimension	1	6	16	96	238
Irr. complex chars.	238	0	0	0	1	239
Irr. rational chars.	2	2	2	2	1	9

Minimal presentations

Permutation degree:	$239$
Transitive degree:	$239$
Rank:	$2$
Inequivalent generating pairs:	$41472$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	238	238	238
Arbitrary	238	238	238

Constructions

Groups of Lie type:	$\AGL(1,239)$
Presentation:	$\langle a, b \mid a^{238}=b^{239}=1, b^{a}=b^{41} \rangle$
Permutation group:	Degree $239$ $\langle(2,239)(3,238)(4,237)(5,236)(6,235)(7,234)(8,233)(9,232)(10,231)(11,230) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 187 & 0 \\ 0 & 78 \end{array}\right), \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{239})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$D_{239}$ $\,\rtimes\,$ $C_{119}$	$C_{239}$ $\,\rtimes\,$ $C_{238}$	$(C_{239}:C_{34})$ $\,\rtimes\,$ $C_7$	$(C_{239}:C_7)$ $\,\rtimes\,$ $C_{34}$	all 7
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut(D_{239})$	$\Aut(C_{239}:C_7)$	$\Aut(C_{239}:C_{14})$	$\Aut(C_{239}:C_{14})$	all 9

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{238} \simeq C_{2} \times C_{7} \times C_{17}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 1682 subgroups in 16 conjugacy classes, 9 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $F_{239}$
Commutator:	$G' \simeq$ $C_{239}$	$G/G' \simeq$ $C_{238}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $F_{239}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{239}$	$G/\operatorname{Fit} \simeq$ $C_{238}$
Radical:	$R \simeq$ $F_{239}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{239}$	$G/\operatorname{soc} \simeq$ $C_{238}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
17-Sylow subgroup:	$P_{ 17 } \simeq$ $C_{17}$
239-Sylow subgroup:	$P_{ 239 } \simeq$ $C_{239}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 56882: $F_{239}$

Order 28441: $C_{239}:C_{119}$

Order 8126: $C_{239}:C_{34}$

Order 4063: $C_{239}:C_{17}$

Order 3346: $C_{239}:C_{14}$

Order 1673: $C_{239}:C_7$

Order 478: $D_{239}$

Order 239: $C_{239}$

Order 238: $C_{238}$

Order 119: $C_{119}$

Order 34: $C_{34}$

Order 17: $C_{17}$

Order 14: $C_{14}$

Order 7: $C_7$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 56882: $F_{239}$

Order 28441: $C_{239}:C_{119}$

Order 8126: $C_{239}:C_{34}$

Order 4063: $C_{239}:C_{17}$

Order 3346: $C_{239}:C_{14}$

Order 1673: $C_{239}:C_7$

Order 478: $D_{239}$

Order 239: $C_{239}$

Order 238: $C_{238}$

Order 119: $C_{119}$

Order 34: $C_{34}$

Order 17: $C_{17}$

Order 14: $C_{14}$

Order 7: $C_7$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 56882: $F_{239}$ ($C_1$)

Order 28441: $C_{239}:C_{119}$ ($C_2$)

Order 8126: $C_{239}:C_{34}$ ($C_7$)

Order 4063: $C_{239}:C_{17}$ ($C_{14}$)

Order 3346: $C_{239}:C_{14}$ ($C_{17}$)

Order 1673: $C_{239}:C_7$ ($C_{34}$)

Order 478: $D_{239}$ ($C_{119}$)

Order 239: $C_{239}$ ($C_{238}$)

Order 1: $C_1$ ($F_{239}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 56882: $F_{239}$ ($C_1$)

Order 28441: $C_{239}:C_{119}$ ($C_2$)

Order 8126: $C_{239}:C_{34}$ ($C_7$)

Order 4063: $C_{239}:C_{17}$ ($C_{14}$)

Order 3346: $C_{239}:C_{14}$ ($C_{17}$)

Order 1673: $C_{239}:C_7$ ($C_{34}$)

Order 478: $D_{239}$ ($C_{119}$)

Order 239: $C_{239}$ ($C_{238}$)

Order 1: $C_1$ ($F_{239}$)

Series

Derived series	$F_{239}$	$\rhd$	$C_{239}$	$\rhd$	$C_1$
Chief series	$F_{239}$	$\rhd$	$C_{239}:C_{119}$	$\rhd$	$C_{239}:C_{17}$	$\rhd$	$C_{239}$	$\rhd$	$C_1$
Lower central series	$F_{239}$	$\rhd$	$C_{239}$
Upper central series	$C_1$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

This group is a maximal quotient of 3 larger groups in the database.

Character theory

Complex character table

See the $239 \times 239$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	7A	14A	17A	34A	119A	238A	239A
	Size	1	239	1434	1434	3824	3824	22944	22944	238
	2 P	1A	1A	7A	7A	17A	17A	119A	119A	239A
	7 P	1A	2A	7A	14A	17A	34A	119A	238A	239A
	17 P	1A	2A	1A	2A	17A	34A	17A	34A	239A
	239 P	1A	2A	7A	14A	1A	2A	7A	14A	239A
56882.18.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
56882.18.1b		$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$
56882.18.1c		$6$	$6$	$-1$	$-1$	$6$	$6$	$-1$	$-1$	$6$
56882.18.1d		$6$	$-6$	$-1$	$1$	$6$	$-6$	$-1$	$1$	$6$
56882.18.1e		$16$	$16$	$16$	$16$	$-1$	$-1$	$-1$	$-1$	$16$
56882.18.1f		$16$	$-16$	$16$	$-16$	$-1$	$1$	$-1$	$1$	$16$
56882.18.1g		$96$	$96$	$-16$	$-16$	$-6$	$-6$	$1$	$1$	$96$
56882.18.1h		$96$	$-96$	$-16$	$16$	$-6$	$6$	$1$	$-1$	$96$
56882.18.238a		$238$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$